for every \(\lbrace (A_k,u_k)\rbrace \subset \mathcal {C}_m\)  and \((A,u)\in \mathcal {C}_m\)  such that \(\mathcal {H}^1(\partial A_k)\)  are equibounded, \(\mathrm {sdist}(\cdot ,\partial A_k)\rightarrow \mathrm {sdist}(\cdot ,\partial A)\) 
locally uniformly in \(\mathbb {R}^2\)  with sdist representing the signed distance function (recall definition at (REF )), and \(u_n\rightarrow u\)  a.e. in \(\mathrm {Int}(A)\cup S\) .
In particular, the restriction to the subfamily \(\mathcal {C}_m\) , which represented already an extension of the more restrictive graph condition assumed in [1]} for the particular setting of epitaxially strained thin films and the starshapedness condition in [2]} for material cavities, was needed in [3]} not only to prove lower semicontinuity, but also to prove compactness with respect to \(\tau _{\mathcal {C}}\) , which fails in \(\mathcal {C}\)  (see Remark REF ).
This enabled in [3]} to prove the existence of minimizers \((A_m,u_m) \in \mathcal {C}_m\)  of \(\mathcal {F}\)  among all configurations in \(\mathcal {C}_m\) , in the following referred to as \(m\) -minimizer of \(\mathcal {F}\) , by means of the direct method of the calculus of variations.
